scholarly journals Uniqueness Problems about Entire Functions with Their Difference Operator Sharing Sets

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Fan Niu ◽  
Jianming Qi ◽  
Zhiyong Zhou
Keyword(s):  
Finite Order ◽  
Difference Operator ◽  
Entire Functions ◽  
Difference Operators ◽  
Transcendental Entire Functions

In this paper, we study the uniqueness questions of finite order transcendental entire functions and their difference operators sharing a set consisting of two distinct entire functions of finite smaller order. Our results in this paper improve the corresponding results from Liu (2009) and Li (2012).

scholarly journals Entire Functions with Separated Zeros and 1-Points

2020 ◽  
Vol 20 (3-4) ◽  
pp. 729-746
Author(s):  
Walter Bergweiler ◽  
Alexandre Eremenko
Keyword(s):  
Finite Order ◽  
Entire Functions ◽  
Specific Form ◽  
Transcendental Entire Functions

AbstractWe consider transcendental entire functions of finite order for which the zeros and 1-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that such functions do not exist at all.

scholarly journals Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order

2011 ◽  
Vol 33 (1) ◽  
pp. 284-302 ◽  
Author(s):  
JÖRN PETER
Keyword(s):  
Finite Order ◽  
Hausdorff Measure ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Gauge Function ◽  
Exponential Map ◽  
Hausdorff Measures ◽  
Transcendental Entire Functions ◽  
Bounded Type Functions

AbstractWe show that the escaping sets and the Julia sets of bounded-type transcendental entire functions of order ρ become ‘smaller’ as ρ→∞. More precisely, their Hausdorff measures are infinite with respect to the gauge function hγ(t)=t2g(1/t)γ, where g is the inverse of a linearizer of some exponential map and γ≥(log ρ(f)+K1)/c, but for ρ large enough, there exists a function fρ of bounded type with order ρ such that the Hausdorff measures of the escaping set and the Julia set of fρ with respect to hγ′ are zero whenever γ′ ≤(log ρ−K2)/c.

scholarly journals Uniqueness of Functions with Its Shifts or Difference Operators

2018 ◽  
Author(s):  
Rajshree Dhar
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Difference Operator ◽  
Difference Operators

It is shown that if a non-constant meromorphic function f(z) is of finite order and shares certain values with its shifts/difference operators then f(z) coincides with that particular shift/difference operator.

scholarly journals Some unity results on entire functions and their difference operators related to 4 CM theorem

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
BaoQin Chen ◽  
Sheng Li
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Entire Functions ◽  
Difference Operators ◽  
Finite Complex

Abstract This paper is to consider the unity results on entire functions sharing two values with their difference operators and to prove some results related to 4 CM theorem. The main result reads as follows: Let $f(z)$ f ( z ) be a nonconstant entire function of finite order, and let $a_{1}$ a 1 , $a_{2}$ a 2 be two distinct finite complex constants. If $f(z)$ f ( z ) and $\Delta _{\eta }^{n}f(z)$ Δ η n f ( z ) share $a_{1}$ a 1 and $a_{2}$ a 2 “CM”, then $f(z)\equiv \Delta _{\eta }^{n} f(z)$ f ( z ) ≡ Δ η n f ( z ) , and hence $f(z)$ f ( z ) and $\Delta _{\eta }^{n}f(z)$ Δ η n f ( z ) share $a_{1}$ a 1 and $a_{2}$ a 2 CM.

scholarly journals Uniqueness Theorems on Difference Monomials of Entire Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Gang Wang ◽  
Deng-li Han ◽  
Zhi-Tao Wen
Keyword(s):  
Finite Order ◽  
Entire Functions ◽  
Uniqueness Theorems ◽  
Complex Constant ◽  
Transcendental Entire Functions ◽  
The Difference

The aim of this paper is to discuss the uniqueness of the difference monomialsfnf(z+c). It assumed thatfandgare transcendental entire functions with finite order andEk)(1,fnf(z+c))=Ek)(1,gng(z+c)), wherecis a nonzero complex constant andn,kare integers. It is proved that if one of the following holds (i)n≥6andk=3, (ii)n≥7andk=2, and (iii)n≥10andk=1, thenfg=t1orf=t2gfor some constantst2andt3which satisfyt2n+1=1andt3n+1=1. It is an improvement of the result of Qi, Yang and Liu.

scholarly journals Uniqueness of Entire Functions concerning Difference Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Chun Wu
Keyword(s):  
Difference Operator ◽  
Entire Functions ◽  
Difference Operators ◽  
Uniqueness Question

We deal with a uniqueness question of entire functions sharing a nonzero value with their difference operators and obtain some results, which improve the results of Qi et al. (2010) and Zhang (2011).

On the fixpoints of composite entire functions of finite order

1994 ◽  
Vol 124 (5) ◽  
pp. 995-1001 ◽  
Author(s):  
J. K. Langley
Keyword(s):  
Rational Function ◽  
Finite Order ◽  
Entire Functions ◽  
Transcendental Entire Functions

Suppose that f and g are transcendental entire functions such that the composition F = f(g) has finite order, and suppose that Q is a nonconstant rational function. We show that N(r, 1/(F – Q)) ≠ o(T(r, F)). The theorem is related to results of Bergweiler, Goldstein and others.

scholarly journals Eremenko’s Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus

2020 ◽  
Author(s):  
D A Nicks ◽  
P J Rippon ◽  
G M Stallard
Keyword(s):  
Finite Order ◽  
Entire Functions ◽  
Minimum Modulus ◽  
Real Zeros ◽  
Modulus Condition ◽  
Transcendental Entire Functions

Abstract We consider the class of real transcendental entire functions $f$ of finite order with only real zeros and show that if the iterated minimum modulus tends to $\infty $, then the escaping set $I(\,f)$ of $f$ has the structure of a spider’s web, in which case Eremenko’s conjecture holds. This minimum modulus condition is much weaker than that used in previous work on Eremenko’s conjecture. For functions in this class, we analyse the possible behaviours of the iterated minimum modulus in relation to the order of the function $f$.

Orlicz lacunary sequence spaces of 𝑙-fractional difference operators

2020 ◽  
Vol 26 (2) ◽  
pp. 173-183
Author(s):  
Kuldip Raj ◽  
Kavita Saini ◽  
Anu Choudhary
Keyword(s):  
Difference Operator ◽  
Lacunary Sequence ◽  
Sequence Spaces ◽  
Difference Operators ◽  
Normed Spaces ◽  
Fractional Difference ◽  
New Approach ◽  
Inclusion Relations ◽  
Difference Sequence Spaces ◽  
Bounded Sequences

AbstractRecently, S. K. Mahato and P. D. Srivastava [A class of sequence spaces defined by 𝑙-fractional difference operator, preprint 2018, http://arxiv.org/abs/1806.10383] studied 𝑙-fractional difference sequence spaces. In this article, we intend to make a new approach to introduce and study some lambda 𝑙-fractional convergent, lambda 𝑙-fractional null and lambda 𝑙-fractional bounded sequences over 𝑛-normed spaces. Various algebraic and topological properties of these newly formed sequence spaces have been explored, and some inclusion relations concerning these spaces are also established. Finally, some characterizations of the newly formed sequence spaces are given.

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